how do I calculate the probability X1 > X2 where X1 and X2 are independent, normally distributed random variables?

Question

how do I calculate the probability X1 > X2 where X1 and X2 are independent, normally distributed random variables?

Answer 1

Stephen,

 

Note that : Pr [x1 > x2] = Pr [ (X1 - X2) > 0].

 

Let X = (X1 - X2) be a new random variable representing the difference of two other random variables

      μ, μ1 & μ2 = the mean values (Mu) for the 3 Normal distributions of X, X1 & X2.

      σ, σ1 & σ2 = the standard deviation values (Sigma) for the 3 Normal distributions of X, X1 & X2.

      Z = the standardized N(0, 1) distribution for which probability tables are given.

 

Given that X1 is N(μ1, σ1), X2 is N(μ2, σ2), and they are independent, X is also normally distributed N(μ, σ),

where μ = (μ1 - μ2), and σ² = (σ1)² + (σ2)^{2  _.}  

Compute these parameters.

 

Then  Z = [X - μ]/σ

 

Look up probability that Z > 0 for your answer.